Is every basis a frame?

Question

It's clear that every orthonormal basis is a frame, but is every basis a frame? In particular, is a non-orthogonal basis always a frame? If yes, why?

I'm asking about general Hilbert spaces, in particular infinite dimensional spaces.

Answer 1

Let $(e_n)_{n\in \mathbb N}$ be an ONB for a Hilbert space. $(\frac{1}{n}e_n)_{n\in \mathbb N}$ is spanning and representation is unique, i.e., it is a basis. But it is not a frame, since $$ \sum_{n\in \mathbb N} \left|\langle e_k, \frac{1}{n}e_n \rangle\right|^2 \leq \frac{1}{k^2}. $$

Answer 2

For finite dimensional vector spaces every basis is a frame in the sense that we can describe all the space in an unique manner by any basis.

For non orthogonal basis we lost the property that

$$ \langle\vec x,\vec v_i \rangle =\vec x\cdot \vec v_i=x_i$$

which holds for orthogonal basis, that is for all $\vec x$ the dot product with the $i^{th}$ basis vector is equal to the component of that vector along the $i^{th}$ coordinate, such that

$$\vec x = \sum_i \langle\vec x,\vec v_i \rangle v_i$$

For infinite dimensional vector spaces refer to the answer by P.Pet.